Rigging a square face to stay coplanar
You want to control the bone roll (Y axis rotation) of the pop-up bones.  This is done most easily with a locked track constraint.  Unfortunately, constraints don't play nicely with IK.  But it's entirely possible to make it work, using an additional bone.
Start by entering edit mode on Armature Popup.  Start by selecting the head of Popup_Center and snapping the cursor to selection (shift s menu for me.)  Now, select Popup_L and duplicate it.  We'll rename that bone to defPopup_L.  Now, with defPopup_L selected, recalculate roll to cursor (ctrl n menu for me.)  Finally, parent defPopup_L to Popup_L.
Change to pose mode.  Delete the (duplicated) IK constraint on defPopup_L.  Give it a locked track constraint, targeting the head of Popup_Center, set to Lock Y axis and Track Z axis (recalculating the roll set the Z axis to point at the cursor.)
Finally, set your mesh object (chiclepopup) to use this new bone instead, by renaming the Popup_L vertex group to defPopup_L.
Repeat this process for Popup_R if you'd like: duplicate it, parent the duplicate to the original, set the roll with a recalculate operation, give it a locked track constraint instead of IK, with appropriate settings, and make sure your mesh is using your new bone for deformation.